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C0-Coarse Geometry of Complements of Z-Sets in the Hilbert Cube

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C0-Coarse Geometry of Complements of Z-Sets in the Hilbert Cube TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 10, October 2008, Pages 5229–5246 S 0002-9947(08)04603-5 Article electronically published on May 20, 2008 C0-COARSE GEOMETRY OF COMPLEMENTS OF Z-SETS IN THE HILBERT CUBE E. CUCHILLO-IBA´ NEZ,˜ J. DYDAK, A. KOYAMA, AND M. A. MORON´ Abstract. Motivated by the Chapman Complement Theorem, we construct an isomorphism between the topological category of compact Z-sets in the Hilbert cube Q and the C0-coarse category of their complements. The C0- coarse morphisms are, in this particular case, intrinsically related to uniformly continuous proper maps. Using that fact we are able to relate in a natu- ral way some of the topological invariants of Z-sets to the geometry of their complements. 1. Introduction In 1972 T. Chapman [3] published what many consider the deepest and most beautiful results in shape theory. The one that stands out is the following Comple- ment Theorem in Shape Theory: Two compact Z-sets in the Hilbert cube have the same shape if and only if their complements are homeomorphic. More generally, he constructed an isomorphism of categories allowing one to describe Borsuk’s Shape Theory in terms of proper maps of complements of Z-sets in the Hilbert cube. Combining both results Chapman proved a rigidity theorem: The complements of two compact Z-sets (or two contractible Q-manifolds admit- ting a boundary [5]) in the Hilbert cube have the same weak proper homotopy type if and only if those complements are homeomorphic. Recently [1] suggested a possibility to describe topological invariants of compact Z-sets of the Hilbert cube in terms of metric-uniform invariants of their comple- ments. In particular two questions were posed there. They were related to the canonical copy of a compact metric space inside its hyperspace with the Hausdorff metric. When the compact space is a non-degenerate Peano continuum those ques- Received by the editors July 17, 2006. 2000 Mathematics Subject Classification. Primary 18B30, 54D35, 54E15; Secondary 54C55, 54E35, 54F45. Key words and phrases. Covering dimension, asymptotic dimension, C0-coarse structure, ANR-space, C0-coarse morphism, uniformly continuous map, compact Z-set, Higson-Roe com- pactification and corona. The first and fourth named authors were supported by the MEC, MTM2006-0825. c 2008 American Mathematical Society Reverts to public domain 28 years from publication 5229 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 5230 E. CUCHILLO-IBA´ NEZ,˜ J. DYDAK, A. KOYAMA, AND M. A. MORON´ tions are about Z-copies in a Hilbert cube. When translated to our current context the questions become: 1. How can the covering dimension of a compact Z-set be described from the outside? 2. What is a manifold from the outside? The above paragraphs provide the main motivation for this paper. Following the spirit of the isomorphism of categories constructed by Chapman in [3] we prove that the topological category for compact Z-sets of the Hilbert cube is isomorphic to the C0-coarse geometry category of their complements. The definition of this C0-coarse geometry category requires a fixed metric on the Hilbert cube, but soon it is pointed out that the definition is, in fact, independent of the chosen (equivalent) metric. The C0-coarse structure of a metric space was introduced by Wright [16] (see also [17]) for different purposes from those pursued herein. We recommend [14] for the basic definitions of coarse spaces. Nevertheless we will recall some of these definitions, if needed. The objects in this category are the complements of Z-sets with the C0-coarse structure induced by a metric in the Hilbert cube. The morphisms are the equiv- alence classes of close coarse maps between them. See [16], [17] and [14] for the basic definitions of C0-coarse maps and C0-close maps. We also obtain our rigidity result. In fact, after the construction of the isomorphism of categories, it is very easy to see that the complements of two compact Z-sets are C0-coarse equivalent if and only if those complements are uniformly homeomorphic (with respect to the metrics induced by the one fixed on the Hilbert cube). In view of the equivalence of categories described above it is natural to consider the problem of describing topological invariants of Z-sets in terms of C0-coarse invariants of their complements. That is how the asymptotic dimension of such a coarse structure on the complement appears as the adequate tool to describe, from the outside, the topological dimension of a Z-set. To do it efficiently we use some results of [7]. In fact we prove that the Hilbert cube is the Higson-Roe com- pactification of the complement of a Z-set for this coarse structure. Moreover, its Higson-Roe corona is exactly that Z-set. That fact allows us to show that the di- mension can be interpreted in terms of the asymptotic dimension of such a coarse space. See [14] and [7] for the definitions of asymptotic dimension, Higson-Roe compactification, and corona for arbitrary coarse categories. So, in some sense, we reinforce the relation between the topological and coarse definitions of dimension and, in particular, we solve the first question of [1]. We also reinterpret the con- cept of external dimension introduced by Mrozik [11] as a characterization of the C0-asymptotic dimension of the complement of a Z-set analogous to that of the topological dimension using extensions of maps to spheres. The concept of asymptotic dimension was defined by Gromov [8] to study as- ymptotic invariants of discrete groups, mainly the fundamental group of manifolds. For finitely generated groups Gromov used the bounded coarse geometry associated to the word metric defined by means of a finite set of generators. The main fact in [8] was that such coarse geometry does not depend on the chosen generating set. Motivated by this, some authors (see for example Dranishnikov [6], Roe [12, 13] and references there) began to develop the asymptotic topology or large-scale ge- ometry. Our paper is related to [6] as we also connect topological invariants of compact spaces to coarse concepts (in our case to C0-coarse concepts). In fact, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use Z-SETS FROM THE OUTSIDE 5231 due to the isomorphism of categories we construct, this paper could help in the program of finding suitable translations of topological concepts to coarse geometry as pursued in [6], because it could be possible to relate all the topological invariants between compacta to the C0-coarse structure of their complements, by means of the categorical isomorphism, and then to adapt them to the general definition of coarse geometry. We do not follow this line herein. On the contrary, due to the fact that, in our case, the coarse morphisms can be represented by proper uniformly continuous functions, we stay as close as possible to Topology in the procedure of translating topological invariants of compact Z-sets to C0-coarse invariants of their complements. In [14], Roe gives a definition of asymptotic dimension for general coarse spaces. Later on Grave [7] proposed other definitions (translating some characterizations from the topological framework). All of them are equivalent in the realm of proper coarse spaces. We complete the paper by reinterpreting the ANR-property (a necessary prop- erty for being a manifold; see the second question above) and the group of auto- homeomorphisms in terms of the C0-coarse structure. We also take this opportunity to describe the homotopical category of compact metric spaces in terms of proper and uniformly continuous maps between their complements. The authors are very grateful to the referee whose detailed suggestions improved the exposition of the paper significantly. 2. Equivalence of categories and applications In this section we establish that the topological category of compacta (the most geometric topological spaces, namely compact and metrizable) can be described in terms of coarse geometry. More precisely, we need to use the so-called C0-coarse geometry introduced in [15] (see also [16]). Later on we translate some topological invariants to the coarse geometry language. As mentioned above we follow [14] for basic definitions and notation in Coarse Geometry. Recall (see [15] or [14, page 22]) that for each metric space (X, d)theC0-coarse structure is the collection E0 of all E ⊂ X × X for which the distance function d, restricted to E, tends to zero at infinity; that is to say, E is controlled if for any ε>0 there is a compact subset K ⊂ X such that d(x, x) <εwhenever (x, x) ∈ E \ K × K. Consider the Hilbert cube (Q, d) with a fixed metric d reflecting its topology. Suppose that X ⊂ Q is a compact Z-set of Q and denote by (Q \ X, E0) the coarse space Q \ X with the C0-coarse structure associated to the metric d |(Q\X)×(Q\X). It is obvious that if X = ∅ is a closed Z-set in Q, then the metric d |(Q\X)×(Q\X) is never a proper metric (recall that a metric is proper when the compact subsets are precisely the closed and bounded sets). On the other hand, there is a concept of proper coarse structure ([14, page 25]) which is important in coarse geometry. To establish that notion first let us recall some definitions from [14]. A coarse structure E on X is a family of subsets E (called controlled sets)of X × X satisfying the following properties: (1) The diagonal ∆ = {(x, x)}x∈X belongs to E.
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